Lep 1.5.15-15

Ultrasonic diffraction at different single and double slit systemsRelated topics
Longitudinal waves, Huygens’ principle, Interference,
1. Record the intensity of an ultrasonic wave diffracted by var-
ious slits and double slits as a function of diffraction angle.
2. Determine the angular positions of the maximum and mini-
Principle
mum values and compare them with the theoretical results.
A plane ultrasonic wave is subjected to diffraction at singleslits of various widths and at various double slits. The intensi-
Set-up and Procedure
ty of the diffracted and interfering partial waves are automati-
Set up the experiment as shown in Fig. 1. Exact adjustment of
cally recorded using a motor-driven, swivel ultrasound detec-
EquipmentAdjustment of the goniometer:
Use the adjusting screws at the back of the mirror and
under its stem to set the mirror by eye to a vertical position
and align it to the zero line of the goniometer table.
Power supply f. ultrasonic unit, 5 VDC, 12 W
Slide the transmitter tightly against the mirror and align it
to the height of the centre of the mirror.
Slide the transmitter back to fit the 16 cm long adjusting
rod in the hole in the centre of the mirror. The rod must
point directly to the middle of the transmitter. Should this
not be the case, again use the adjusting screws to re-
adjust the mirror. Remove the rod so that the transmitter
can be brought to the focal point of the mirror. The dis-
tance from the centre of the mirror must be exactly
PHYWE series of publications • Laboratory Experiments • Physics • PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen
Ultrasonic diffraction at different single and double slit systems
To adjust the height of the receiver, turn it with its swing
arm as near as possible to the mirror. It might be neces-
Faulty intensity modulation may occur in spectra as a result of
sary here to first unlock the transport stop (to do this, pull
interference in the measurement field. To keep such interfer-
the yellow screw of the swing arm beneath the goniometer
ence as small as possible, do not carry out experiments in too
table). Bring the receiver to the same height as the trans-
narrow rooms or in the direct vicinity of reflecting surfaces
(walls, cupboards etc.). It is recommended that the measuring
Set the receiver swing arm to zero. The axis of the receiv-
and supply instruments be installed behind the mirror if possi-
er must correspond with the goniometer zero line.
ble. Further to this, the person carrying out the experiment
Bring the receiver to the end of the swing arm.
should not stand too close to the measurement field.
When adjustment is properly made, the axes of the mirror,
Should asymmetries occur in the intensities in spectra, these
transmitter and receiver must be on a common line and
can as a rule be avoided by slightly turning the object holder
this must be exactly above the zero line of the goniometer
around the 90° line on the goniometer table.
Fit the object holder with its centering pin in the central
Theory and Evaluation
socket of the goniometer table, with the feet of the holderpointing to the mirror. Position the holder on the 90° line on
When a wave hits a slit then, acc. to Huygens’ principle,
spherical waves are emanated from each point of the slit
To prepare the slit and double slit, carefully insert the cor-
opening. The individual partial waves interfere with each other
responding metal sheets in the guide grooves of the object
behind the obstacle. According to their phase position, they
holder, then align these exactly centrally symmetrical with
intensify each other in certain directions, or extinguish each
the zero line of the goniometer table.
other. In the direction of the incident waves (w = 0) (see Fig. 2),
To avoid interfering sound reflections, use the carrier foam
all partial waves have the same phase and intensify each
as wave absorbent. Place it between the object holder and
mirror, tightly against the holder and with its opening sym-metrically towards the diffraction object.Connection of instruments:
Connect the transmitter to the diode socket of the ultrasonicunit that is marked TR1, and operate it in “Con“ continuousmode. Connect the receiver to the left BNC socket (prior to theamplifier). Further, use the BNC cable to connect the analogoutput of the ultrasonic unit with the input of the control unit(pay attention to the polarity of the adapter), and the latter unitto the PC by means of the RS 232 data cable.For control of the goniometer, connect the socket underneaththe goniometer plate with the control unit.With the “Cal“ key of the control unit pressed (release of themotor drive) position the swing arm at 0°. Following this,deactivate the “Cal“ function.Use the software to set the range of swing of the receiver to±50°.
Fig. 2: Diagram of diffraction at a slit.
To ensure proportionality between the input signal of thereceiver and its analog output signal, avoid operating theultrasonic unit amplifier in the saturation range. Should such acase occur and the “OVL“ diode light up, reduce either the
With sound waves, the sound pressure p(w) is represented as
transmitter amplitude or the input amplification of the receiv-
a function of the diffraction angle w by the so-called slit func-
er. It is purposeful here to adjust the amplification at the zero
position of the receiver so that the “OVL“ diode just no longerlights up.
5 Measurements are described in the following.
Interference pattern of a slit of width b = 6 cm.
Where b is the slit width, l the wavelength of the sound and p
Interference pattern of a slit of width b = 4 cm.
the alternating sound pressure that is recorded by the sound
Interference pattern of a double slit of slit width b = 2.5 cm
Equation (1) is also valid, in squared form, for transversal elec-
tromagnetic waves (optics), as in this case the intensity is
given by the square of the amplitude.
Set the measurement mode to repeat measurement.
For w = 0, an indefinite expression is obtained, as both the
Interference pattern of a double slit of slit width b = 2.0 cm
numerator and the denominator are null. Application of the
and slit separation s = 5.0 cm. Subsequently repeat mea-
l’Hospital rule shows, however, that for w = 0, the quotient
surement with a single slit of width b = 2.0.
assumes the value 1. Zero positions are at sin u = 0, i.e. at
PHYWE series of publications • Laboratory Experiments • Physics • PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen
Ultrasonic diffraction at different single and double slit systems
u = np (n = 1, 2, 3,.). From this, the positions of the minima
As the central maximum does not generally lie exactly at 0°, it
is purposeful to determine the angular distance 2 w betweentwo extremes lying symmetrically to the zero line. Table 1 listsboth 2
w angle of extremes determined from Fig. 3 and Fig. 4,
as well as the wavelength values l calculated using equations
The numerator in equation (1) becomes 1 when u = 1/2(2n+1)l,i.e. when u is an uneven multiple of p/2. Intensity maxima
Table 1: Evaluation of the interference curves shown in Figs. 3
From equation (1) it also follows for the ratio of the intensities
Fig. 3 and Fig. 4 each show interference patterns of ultrason-ic waves at single slits, but with different split widths b. The
two curves were recorded using the same transmitter perfor-
The mean value of the wavelength values listed in Table 1 is:l = (0.862±0.034) cm.The transmitter operates at a frequency of f = 40 kHz. Fromc = f · l (c = 343.4 ms-1 at T = 20°C) it follows that, in com-plete agreement with the experiment, l = 0.858 cm.The curves in Figs. 3 and 4 also show that, at a constant emit-ter performance, the intensity of the maximum of zero orderbecomes less when the slit width is reduced. For the corre-
Fig. 3: Interference pattern of ultrasonic waves diffracted at a
Fig. 5 shows a diagram of diffraction at a double slit. The slitwidth is again given by d and the distance apart of homolo-gous partial waves by the slit separation s.
Fig. 4: Interference pattern of ultrasonic waves diffracted at a
Fig. 5: Diagram of diffraction at a double slit.
PHYWE series of publications • Laboratory Experiments • Physics • PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen
Ultrasonic diffraction at different single and double slit systems
With a double slit, the intensity distribution is given by:
Table 2: Evaluation of the interference curve shown in Fig. 6.Double split s = 5.5 cm and b = 2.5 cm.
The slit function is now additionally modulated by a cos func-
tion. The minima of the individual slits (1st class minima) arestill as previously. Additional 2nd class minima occur, howev-
er, namely there where the cos function is null. This is always
Additional 2nd class maxima occur when the cos factor is 1,
With a little skill, the dimensions of the double slit can now be
so chosen, that the 1st minimum of the single slit (1st classminimum) coincides with a minimum of the double slit (2nd
class minimum). This is the case, for example, when s = 5/2 b,
as the 1st minimum of the single slit then coincides with the
3rd minimum of the double slit. Fig. 7 shows such a case, with
Fig. 6 shows the interference pattern of a double slit having
the recording of the double slit curve superimposed on the
Table 2 shows the corresponding evaluation and the values forl calculated from equations 5 and 6. A mean value of l =
It can be seen that the single slit curve envelopes the double
(0.848±0.035) cm is obtained. In Fig. 6, at w
indentation is to be seen, that is concordant to the position ofthe first minimum of the single slit.
Fig. 6: Interference pattern of ultrasonic waves diffracted at a
Fig. 7: Interference pattern of ultrasonic waves diffracted at a
double slit of s = 5.5 cm and b = 2.5 cm.
double slit of s = 5.0 cm and b = 2.0 cm. (The single slitcurve always envelopes the double split curve)
PHYWE series of publications • Laboratory Experiments • Physics • PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen